The women no need to wear until like that. Dubai is strangely much more progressive compared to Malaysia or Brunei!
Just that as a woman you have to know your place. Let me tell you their men's attitudes.
We setup a our booth the day before. My manager is a lady. We come back the next day, it is all in a mess. So we started to clean things up but there is this unsightly item in the way. This burly looking Arab guy comes to our booth and tells us he is the organizer, the exhibition opens in a few minutes and he wants this unsightly item out of the way. My manager replied saying that we have set things up well yesterday and apparently "you people messed it up".
Guess how the guy replied?
"DON'T SAY I MESS IT UP! YOU REMOVE THIS ITEM BEFORE EVERYTHING STARTS!"
She suddenly realized she is a very "patriarchal" society. You see she said "you people" but this guy takes it as he is being the one singled out and he being THE MAN, cannot be faulted by a woman and hence his attitude. She kept quiet to let him have his way.
In public a man cannot be seen taking orders or faulted by a woman over in that part of the world.
Hope this doesn't scare you women from going there. Just meekly back away from confrontation with men.
As for their men's behavior, they can get pretty overt in their gawking. There was a group of men checking out the 4ss of one of the women in my party just because she was wearing tight jeans. Patriarchy oh patriarchy!
If you are visiting the Grand Mosque in Abu Dhabi, there are abayas for rent. The security guards will see if your clothing is fit to enter.
As for transport, taxis are everywhere. Hail one if you need one. Also check out the Dubai Metro. The stations are like space age stuff you see in Star Trek. Very affordable and brings you to the city center as in Deira.
Accommodation is not cheap here but is the cheapest in June because it is the hottest. As for clothes it can get cold at night especially in the desert if you opt for the desert tour. Jeans and T-shirts work well.